Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption

TL;DR

This paper establishes the global existence of classical solutions to a chemotaxis model with singular sensitivity and signal absorption, extending previous results to broader parameter ranges and initial data, including the one-dimensional case.

Contribution

It proves the existence and boundedness of solutions for a generalized chemotaxis system with singular sensitivity, covering a wider parameter space and initial conditions.

Findings

Global existence of solutions for the chemotaxis system under specified conditions.

Boundedness of solutions in the one-dimensional case for all positive parameters.

Extension of classical solutions results to more general parameter regimes.

Abstract

Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v} \nabla v ) + \kappa u -\mu u^2\\ v_t &= \Delta v - u v \end{split} \] in a bounded domain $\Omega \subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $\chi>0$, $\mu>0$, $\kappa\ge 0$.